Definition:Conjugate Point/Dependent on N Functions

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Definition

Let $K$ be a functional such that:

$\displaystyle K\sqbrk h=\int_a^b \paren {\mathbf h'\mathbf P\mathbf h'+\mathbf h\mathbf Q\mathbf h}\rd x$

Consider Euler's equation related to the functional $K$:

$-\dfrac \d {\d x} \paren {\mathbf P\mathbf h'}+\mathbf Q\mathbf h=0$

where $\mathbf P$ and $\mathbf Q$ are symmetric matrices.

Let the set of solutions to this equation be

$\lbrace {\mathbf h}^{\paren i}=\paren {\sequence {h_{ij} } }:i,j\in\N_{\le N}\rbrace$

Suppose

$\exists j:\forall k\ne j:\paren {\map {\mathbf h^{\paren j} } a=0}\land \paren {\map {h_{jj}'} a=1,h'_{jk}=0}$

Let the determinant, built from $h_{ij}$, be such that:

$\size {h_{ij} }\paren{\tilde a}=0$

Here $i$ denotes rows, and $j$ denotes columns.


Then $\tilde a$ is said to be conjugate to point $a$ with respect to the functional $K$.


Sources

1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.29$: Generalization to n Unknown Functions